Written by Ion Saliu on February 17, 2002.
I come back to the idea presented in a previous post: Dynamic versus Static in Lotto, Lottery Analysis. I ran my lottery software Combinations.exe to generate ranges of combinations around the field's midpoint or around the probability median for a 6/69 lotto game.
I applied the test to PA 6/69 lotto game (a very tough game, indeed; the jackpot odds are '1 in 119,877,472'). I generated first 1000 combinations around the mid-point index: 59,938,736. That is, the lotto program generated combinations between the counts of 59,938,236 and 59,939,236. Next, I checked for winners in the output file against real draws in PA 6/69 lotto (350 real lottery draws at my discretion). The 'mid-point output' achieved 3 '5 of 6' hits and 140 '4 of 6 hits'. The game odds are '1 in 317,000' for '5 of 6' and '1 in 4092' for '4 of 6'. The '5 of 6' hits were recorded in 350 x 1000 = 350,000 combinations. Therefore, the 'mid-point' strategy was about 3 times better than random play in the '5 of 6' case. For the '4 of 6' case, '1 in 4092' applied to 350,000 lotto combinations translates into 85. The mid-point strategy hit 140 times, or about one and a half times better.
The next test was applied against the 'FFG median', instead of the mid-point. The FFG median is calculated in FFG for a degree of certainty equal to 50%. The FFG median is 83,092,731 for a 6/69 lotto game. A number of 1000 combinations were generated in the 83,092,231 83,093,231 range. The results were undoubtedly better. The strategy yielded 6 '5 of 6' hits and 238 '4 of 6' hits. Compared to purely random play, the 'FFG median' strategy was 6 times better for '5 of 6' and about 3 times better for '4 of 6'.
I saw in the newsgroups similar tests applied to UK 6/49 lotto game, with lower odds than PA's 6/69 monster lotto. This time, the results were inverted: the mid-point strategy was favored! It became clear to me that this type of lottery strategy favors the probability (FFG) median in high odds games. The higher the odds, the better the FFG median strategy fares.
I did the test for 10000 lotto combinations against the same 350-draw data file (PA 6/69 lotto). I ran the test for both the midpoint and FFG median: 5000 below and 5000 above (actually, there are 10,001 lotto combinations, including the pivots). The results were dramatic!
1) The midpoint strategy performed worse than in the 1000 test; even worse than random play! It recorded 6 '5 of 6' hits and 442 '4 of 6' wins. That's about half of what lotto random play yields!
2) The 'probability median' strategy was far better than the '1000-combination test' and far-far better than random play.
- 1 '6 of 6' (jackpot) win (34 times better than random play!);
- 75 '5 of 6' hits;
- 2,232 '4 of 6' wins!
Again, a random play for an equivalent amount of combinations (350 x 10001 = 3,500,350) should yield:
- 0.029 '6 of 6' hits;
- 11 '5 of 6' hits;
- 855 '4 of 6' wins.
I compared these results with the results in a 6/49 game (UK, 642 draws).
The 10,001 runs did not yield jackpot wins. The lotto jackpot hits occurred at 50,001-combination runs!
1) Midpoint strategy:
- 27,439 hits (1 '6 of 6')
2) FFG median strategy:
- 34,616 hits (2 '6 of 6').
Now, the FFG median strategy was favored over the midpoint strategy!
It is very clear that this type of play favors the high odds games lotto by far. The lottery strategy is insignificant for lower odds games, such as pick-3 or pick-4 lotteries. I don't have a formula for it. I don't think this type of strategy has an under-pinning formula. The explanation relies on the type of combinations in various areas of the lotto field. The combinations in the FFG median area are largely part of the bell distribution. On the other hand, the purely random field includes many 'hardly-to-come-out-in-a-lifetime' combinations. Combinations like 1-2-3-4-5-6 and the like, or 44-45-46-47-48-49 and the like. Yes, some dudes will jump off: "They have an equal probability to come out!" But they won't come out. One can bet the entire budget surplus of the U.S. treasury in 2000 on lotto combinations such as the above. The results would be the red figures of the year 2002. I can give here a summary explanation, other than Markov chains. The lotto balls would need to be mixed for millions of hours before any drawing. Then, 'weird' combinations would have a reasonable chance to come out.
Probability is viewed by many as a static phenomenon. "The individual probability in the pick-3 lottery is 1/1000. Therefore every lottery combination has an equal chance of appearance." That's correct, insofar as no dynamic processes are involved. But the lottery is a dynamic process. There is more to the picture than meets the eye. The formula of individual probability becomes just a part of inclusive systems (formulae). I have found that many random processes are ruled by the Fundamental Formula of Gambling (FFG). Equal individual probabilities do not lead to equal frequencies. The pick-3 combinations do not appear equally. There are 1000 possibilities in the pick-3 game. In 1000 tries, each and every combination should come out, right? NOT! FFG shows that only around 63 percent of the pick-3 combinations come out in 1000 draws, any 1000 consecutive draws. Some 37% of the pick-3 combination need more than 1000 draws to come out: 1500 draws, 3000 draws, even 6000 lottery drawings!
Let's answer a very interesting question. What is the probability that ALL pick-3 combinations will come out in 1000 lottery drawings (tries)?
Let's start with the simplest gamble: coin tossing. The probability is ½. Then, what is the probability that 'heads' and 'tails' will come out in 2 tosses? We apply the 'Binomial Distribution Formula'. You can use my SuperFormula.exe to do the calculations. The probability for either 'heads' or 'tails' to come out EXACTLY once in 2 tries is: ½ or 50% (no surprise there). We must have EXACTLY 1 'heads' and 1 'tails' in 2 tosses. That is, ½ x ½ = Ό (25%). The chance to get EXACTLY 1 'heads' and 1 'tails' in 2 tosses is '1 in 4'. The event will be 'heads-tails' or 'tails-heads'. It means that if we toss the coin 2 times, and do such a thing 4 times, we can expect to get a result such as 'heads-tails' or 'tails-heads'. Sometimes, we may get it in the first try of 2 tosses; other times, we may have to wait 10 or more 2-toss tries.
Let's take a higher odds game: rolling the dice. A cube has 6 point faces. What is the probability to get ALL 6 point faces in 6 rolls? The individual probability is 1/6. We apply again the 'Binomial Distribution Formula'. The probability to get EXACTLY 1 face point in 6 tries is 0.4 (40%). That chance is equal for each point face. The final result for ALL 6 point faces in 6 rolls is 0.4 to the power of 6: 0.4 ^ 6 = 0.004, or '1 in 250'. We need to repeat the 6-roll event 250 times to get 'ALL 6 point faces in 6 rolls'. Sometimes it can happen in 10 or 100 tries; other times, it may take 500 or 1000 or more '6-roll' tries.
The pick-3 results enter the astronomical realm. We can't even say those numbers. The individual probability is 1/1000. We apply again the 'Binomial Distribution Formula'. The probability to get EXACTLY 1 pick-3 combonation in 1000 lottery drawings is 0.368 (36.8%). It is a pretty high probability. That chance is equal for each pick-3 combination. The final result for ALL 1000 pick-3 lottery combinations in 1000 drawings is 0.368 to the power of 1000: 0.368 ^ 1000 = 'my-calculator-can't-compute-that-operation number'! Raising only to the power of 100 gives 0.'44 zeros'3844! Forget about it! There is a far, far, far better chance for the Earth colliding with an asteroid that would put an end to all life.
Undoubtedly, the lotto combinations have different frequencies as a matter of truth. Some repeat, some repeat more often, some do not appear at all in various ranges of trials. The process is not blind. It can be analyzed scientifically. As far as I am concerned, the Fundamental Formula of Gambling (FFG) has provided the best analytical tools. Its fundamental discovery: In 50% of the cases, each combination repeats after a number of trials less than/equal to the probability median of the game. That's a law. It is validated in any game of chance, lottery included. I analyzed the facts in other posts: Comparative lottery lotto software programs .
I showed that in a lotto 6/48 (or 6/49) game, every number will repeat within 6 drawings in over 50% of the situations. Let's dig deeper into this statement. So, the probability of a lotto 6/49 number to repeat is ½ times 1/6 (the number will repeat after 1, or 2, or 3, or 4, or 5, or 6 drawings). The combined probability is ½ x 1/6 = 1/12. Actually, it is .54 * 1/6 = 0.09. But the lottery draws 6 numbers, therefore the probability to get all 6 numbers repeating after 1 to 6 drawings is: 0.9 ^ 6 = 0.000000531 (raising to the power of 6). The result can be expressed also as '5.31 times in 10 million', or '1,881,676 to 1'. Well, the odds are better than playing 6 random numbers (1 in 13,983,816): 7 and half times better. That's a law, too. No matter when you play such a lotto strategy, your odds will always be seven times better than purely random play.
You can have my free lotto software (LotWon or MDIEditor and Lotto WE) create the so-called skip charts. Simply selecting 6 lotto numbers that show as current skips figures between 0 and 5 improves your lotto 6/49 odds seven-fold! There is no doubt or mystery here. It is a formula-backed fact!
As of the 'midpoint' and 'FFG median' strategies presented in the beginning: I do not endorse them. I have no formula to lay a foundation. They may be expensive as well. They certainly need further lotto filtering, using LotWon filters. You should never-ever apply ordinary 'cigarette-butt filters'!
Best of luck!
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